Device for generating arcs



June 1968 D. s. CHISHOLM DEVICE FOR GENERATING ARCS 5 Sheets-Sheet 1 Filed July 28, 1967 INVENTDR. Doug/0.5 .5. C/HlS/70/rn June 11, 1968 D. s. CHISHOLM 3,387,493

DEVICE FOR GENERATING ARCS Filed July 28, 1967 3 Sheets-Sheet 'l INVENTOR. Oouy/as 3 Ch/lsho/m HGEN T June 11, 1968 s, cHlSHOLM 3,387,498

DEVICE FOR GENERATING ARCS Filed July 28, 1967 6 Sheets-Sheet 3 Q is Q I k INVENTOR.

Doug/a5 S. C/nlsho /m FYGENT United States Patent 3,387,498 DEVICE FOR GENERATING ARCS Douglas S. Chisholm, Midland, Mich., assignor to The Dow Chemical Company, Midland, Mich., a corporation of Delaware Continuation-impart of application Ser. No. 507,269,

Nov. 12, 1965. This application July 28, 1967, Ser.

14 Claims. (Cl. 74-1) ABSTRACT on THE DISCLOSURE An apparatus is described to trace a circular arc without access to the center of the circle employing a reference triangle at least one of Whose sides is parallel to a reference radius of a circle.

This application is a continuation-in-part of my c0pending application Ser. No. 507,269, filed Nov. 12, 1965, and now abandoned.

This invention relates to a device for the generation of arcs and spherical surfaces, and more particularly relates to a device for the generation of circular arcs and spherical surfaces which permits the use of relatively compact equipment.

In many applications, it is desired to trace an arc of a circle which has a radius of relatively great size and the center of the circle is inaccessible or inconveniently accessible. Oftentimes in drafting or drawing, arcs of circles are required which have a radius sufficiently large to prohibit drawing of such an arc conveniently by conventional means. Oftentimes, it is desired to construct buildings having a spherical surface, such buildings being a segment of a sphere wherein access to the center of the sphere is not readily available. By the term spherical surface is meant any surface which is a portion of the surface of a sphere. Methods of constructing buildings utilizing brick-laying machines or plastics, such as are described in Belgian Patent 612,405, in essence incrementally form a structure by the sequential laying of bricks or deposition of foam plastic elements. Oftentimes, it is undesirable to construct such buildings in the form of a hemisphere employing a centrally pivoted boom or arm of fixed length. Frequently, in order to minimize the head space and make use of the maximum floor area, it is desirable to construct such buildings in the form of a segment of a sphere.

It would be advantageous if there were available an apparatus which would permit the construction of an arc of a circle without access to the center.

Further, it would be advantageous to have an apparatus for tracing a spherical surface without access to the center of the sphere.

Furthermore, it would be beneficial if there were available a simple apparatus for the generation of an arc of a circle and spherical surfaces which is uncomplicated and is readily employed to generate circles or spherical surfaces of varying radii.

These benefits and other advantages in accordance with the present invention are achieved in an apparatus for the generation of circular arcs without access to the center of the arc, the apparatus comprising in cooperative combination a first arm, the first arm having a first end and a second end, the second end of the first arm being pivotally affixed to a base, the base adapted to be positioned generally on a radius of the desired are, a second arm and a third arm in cooperative adjustable combination with the first arm defining a reference triangle, means to maintain the second arm parallel to a lice radius of the desired circle passing through the base and means to maintain the third arm parallel to a radius passing through the are being traced, the third arm being disposed generally between the second arm and the second end of the first arm.

Further features and advantages of the present invention will become more apparent from the following specification taken in connection with the drawings wherein:

FIGURE 1 schematically depicts an elevation view of one embodiment of an apparatus in accordance with the invention.

FIGURE 2 depicts an elevation view of an alternate embodiment of the invention.

FIGURES 3 and 4 also depict alternate embodiments of the invention. 1

FIGURE 5 depicts the geometrical relations in the embodiments of FIGURES 3 and 4.

FIGURE 6 depicts the geometry of operation of the embodiments of FIGURES 1 and 2.

In FIGURE 1 there is illustrated an elevati-on'al view of an apparatus generally designated by the reference numeral 10 for the generation of circular arcs and spherical surfaces without access to the center of such a circle or surface. The apparatus 10 comprises in cooperative combination a base 11. The base 11 comp-rises an arm support 12 and a fixed support 13. The arm support 12 is pivotally connected to the fixed support 13 by means of a pivot 14. The arm support 12 has disposed therein a pivot means or shaft 16 lying on a radius r of the circular arc to be generated. A first arm 18 having a first end 19 and a second end 20 is pivotally attached to the arm support 12 by means of the pivot 16. The arm 18 defines a generally longitudinal slot 22 within the arm. A second arm 24 is slidably affixed to the first arm 18 and is so constructed and arranged that the arm 24 travels generally parallel to the arm 18 (in effect forming an adjustable length arm). A pair of support members 26 and 27 is rigidly aiiixed to the arm 24. The members 26 and 27 define therein internal cavities 28 and 29 adapted to receive at least a portion of the arm 18. The second arm 24 has a first end 30 to which the supports 26 and 27 are rigidly affixed. The arm 24 has a second or operating end 31 remotely disposed fr-om the first end 19 0f the first arm 18. A third arm 35 having a first end 36 and a second end 37 is pivotally affixed to the first arm 18 by means of the pivot 38. A fourth arm 40 has a first end 41 and a sec- 0nd end 42. The first end 41 of the fourth arm 40 is pivotally attached to the second end 37 of the third arm 35. A support 44 is slidably positioned over the arm 18 and disposed between the pivot 38 and the pivot 16. The support 44 is pivotally attached to the second end 42 of the fourth arm 40 by means of the pivot 45. A motion multiplying means generally designated by the reference numeral 48 is in operative association with the fourth arm 40. The motion multiplying means 48 comprises a rack 50 having a plurality of teeth 51. A spur gear 52 is disposed on a rotatable shaft 53 rotatably positioned within the second end 20 of the first arm 18. A drum pulley or gear 55 is disposed on the shaft 53 and is rigidly mounted relative to the gear 52. An idler pulley or gear 58 is rotatably mounted generally adjacent the second end 20 of the first arm 18 and adapted to rotate in a plane containing the pulley or gear 55'. A motion transmitting means 60, such as a cable or chain, is disposed in operative engagement with the pulley or gear 55 and the idler 58. The motion transmitting means 60 is secured to the second arm 24 generally adjacent its first end 30. A second arm positioning means 62 is in operative engagement with the idler 58. The second arm positioning means comprises a motor 63 of the reversible type, a worm 65 secured to the output shaft of the motor and a worm gear 66 in operative engagement with the worm 65 and adapted to rotate the pulley 58. A switch 68 of the single-pole double-throw variety is adjustably supported on the third arm 35 by means of the adjustable switch support 69. A power source 71 is in operative communica ion with the motor 63 by means of the lines or conductors 72, 73, 74, 75 as well as the switch 68. The switch 68 comprises a switch arm 76, a common pole 77, a first pole 7S and a second pole 79. The switch arm 76 is pivoted and the pole 77 adapted to make contact with the first pole 78 or the second pole 79 when displaced from a vertical position. The arm 40 has a length between the pivots disposed at the first and second ends of r/a and the ratio of the diameter of the spur gear 52 to the diameter of the gear 55 is a. As the arms 18 and 24 are raised, i.e., moved toward the vertical or reference radius passing through the base, the arm 24 is driven toward the pivot 16. A radius of a circle which is traced by the apparatus is illustrated as a radius r passing through the pivot 16. A portion of the radius of the generated circle indicated as r is depicted adjacent the second end 31 of the second arm 24.

In operation of the apparatus as illustrated in FIG- URE 1, the base 11 is disposed in such a manner that the pivoting point 16 lies on a radius of the desired circle to be generated. The third arm 35 is positioned to be vertical by virtue of the activation of the motor 63 which extends the second arm 24 by means of the motion transmitting means 60 while rotating the pulley or gear 55 which in turn activates the spur gear 52 causing the second end 42 of the fourth arm 40 to be slidably positioned on the first arm until the third arm is in the vertical position. Any deviation of the third arm from the vertical position results in displacement of the second arm in such a manner that its second end traces an arc of a circle having a radius equal to ka, wherein it is the length between pivots of the fourth arm and a is the ratio between the pitch diameter of the spur gear 52 and the pitch diameter of the gear 55.

Beneficially, almost any desired operating element may be positioned at the second end 31 of the second arm 24. A scriber, pen or pencil may be employed if a circular arc is to be transferred to a suitable substrate, or a cutting means, such as a cutting torch, if it is desired to cut metals. An operating head such as is disclosed in Belgian Patent 612,405 is readily employed for the construction of buildings and enclosures having the general form of a spherical zone or segment wherein the apparatus is rotated about the pivot 14 on the support 13 as the structure is prepared. Alternately, the rotation of the first arm 18 about the pivot 16 may be accomplished by any desired external force if the assembly is selfsupporting.

In FIGURE 2 there is illustrated an alternate embodiment of the invention generally designated by the reference numeral 90. The apparatus 90 comprises a reference arm 91 having a first end 92 and a second end 93, a first arm 95 having a first end 96 and a second end 97, the first end 96 of the first arm 95 being pivotally attached to the first end 92 of the reference arm 91 at the pivot point 98. A second arm 99 is slidably afiixed to the first arm 95 by means not shown. The second arm 99 has a first end 100 and a second end 101. An indicating means 102 is disposed within the second end 101 of the arm 99. A third arm 103 is pivotally afiixed to the first arm 95 at a location intermediate between the first and second ends thereof. The third arm 103 has a first end 104 and a second end 105. The first end 104 is pivotally attached to the first arm 95 by means of the pivot 106. A fourth arm 107 having a first end 108 and a second end 109 is pivotally afi'ixed to the arm 103 generally adjacent its second end 105. The second end 109 of the fourth arm 107 is slidably secured to the first arm 95 generally adjacent a location lying between the pivot 106 and the pivot 98. A fifth arm 110 having a first end 111 and a second end 112 is pivotally affixed to the second end of the third arm 103. The second end 112 of the fifth arm is pivotally secured to the reference arm 91 by means of the pivot 113. The distance between the pivots 113 and 98 is equal to the distance between the pivot disposed at the end of the third arm 103 and the pivot 113, and the distance between the pivot 113 and the pivot disposed at the first end 111 of the fifth arm 110 is equal to the distance between the pivot 98 and the pivot 106, thereby forming a parallelogram. A rack 115 is operatively affixed to the second end 109 of the fourth arm 107. A spur gear 118 is rotatably disposed within the second end 97 of the first arm 95. The spur gear is rigidly affixed to a drum or pulley 120. The ratio of the pitch diameter of the drum 120 and the spur gear 118 is equal to a. A motion transmitting means 121, such as a cable, passes over an idler pulley 122 disposed generally adjacent the pivot 98 on the first end 96 of the first arm 95. The cable 121 is rigidly secured to the second arm 99.

In operation of the apparatus of FIGURE 2, the reference arm 91 is disposed along a radius of a circle to be scribed. The first arm 95 is then pivoted about the pivot point 98. The third arm 103 is maintained parallel to the reference arm 91 by the four-bar linkage formed from the reference arm, the first arm, the fourth arm and the fifth arm. The fourth arm 107 slidably afiixed to the first arm 95 is moved along the first arm 95 displacing the rack 115 and causing rotation of the spur gear 118 and the pulley 120 causing the cable 121 to move the second arm 99 and the indicating means or operating means 102 to scribe an arc of a circle. As the first arm 95 is pivoted toward the reference arm 91, the indicating means 102 is moved toward the pivot 98. The fourth arm 107 has a length between the pivoted ends 108 and 109 of r/ a wherein r is the radius of the arc of the circle and a is the ratio between the movement of the rack 115 and movement of the cable 121, a being greater than unity. The embodiment of FIGURE 2 is particularly adapted and suited for applications such as in drafting and the like wherein a relatively small portable unit is desired, whereas the embodiment of FIGURE 1 is particularly suited for larger scale applications such as building, construction and the like.

In FIGURE 3 there is depicted an alternate embodiment of the invention generally designated by the reference numeral 130. The apparatus comprises in cooperative combination a base 131 having pivotally afiixed thereto an arm 132. The arm 132 has a first end 133 and a second end 134. The first end 133 is secured to the base 131 by a pivot permitting rotation of the second end 134 about the pivot 135 in a solid angle, i.e., in the plane of the illustration as well as in a plane normal to the plane of the paper. A second arm 136 is slidably affixed to the arm 132. The arm 136 has a first end 137 and a second end 138. A bearing 139 is slidably affixed to the arm 132 in such a manner as to permit movement of the bearing 139 along the length of the arm 132. The first end 137 of the second arm 136 is pivotally aifixed to the bearing 139 by means of a pivot 140. A third arm 141 having a first end 142 and a second end 143 is pivotally affixed to the second end 138 of the second arm 136 by means of a pivot 144. A bearing or support means 145 is slidably affixed to the arm 132 at a location between the bearing 139 and the first end 133. The second end 143 of the arm 141 is pivotally afiixed to the bearing 145 by means of the pivot 146. A shaft 149 is rigidly secured to the second end 134 of the first arm 132. The shaft has rotatably afiixed thereto a first idler pulley or gear 151 and a second idler gear 152. A second shaft 155 is rotatably afiixed to the second end 133 of the arm 132 and has rigidly afiixed thereto a first driven pulley or gear 156 and a second driven pulley or gear 157. A first linear motion transmitting means or sprocket chain 158 passes about the driven gears 156 and 151 in the form of an endless loop and a second linear motion transmitting means 159 passes about the gears 152 and 157 also in the form of an endless loop. The chain 158 is aflixed to the bearing 139 andthe chain 159 is afiixed to the bearing 145 in such a manner that upon rotation of the gears 156 and 157, the bearings 145 and 139 travel in a like direction and at different linear speeds dependent on the ratio between the driven gears 156 and 157. The second arm136 has afiixed thereto a means 160 to maintain the second arm 136 parallel to a radius r of a circle to be traced. Beneficially, the means 160 may be a gravity operated switch such as the switch and motor arrangement employed in the apparatus of'FIGUR-E 1. In cooperative combination with the means 160 by means of connections not shown is a motor 162 in operative engagement with a gear 163 which rotates the shaft 155 and the ears 156 and 157. As the second end 134 of the first arm 132 is pivoted in the plane of the paper about the pivot 135 and the second arm 136 is maintained parallel to theradius r, the pivot points 140, 146 and 144 de scribe arcs of circles having their centers lying at a location below the base 131.

In FIGURE 4 there is depicted an alternate modification of the apparatus of FIGURE 3 generally designated by the reference numeral 165. The apparatus of FIGURE 3 is depicted in what can be considered an elev ational view. FIGURE 4 is a plan View of this apparatus depicting an alternate mode of imparting motion to the second and third arms. The apparatus 165 comprises a first arm 166 having a first end 167 and a second end 168. A first bearing 169 is slidably affixed to the first arm 166 generally adjacent the second end 168. A second bearing 170 is slidably affixed to the arm 166 generally adjacent the first end 167. A second arm 171 is pivotally aifixed to the bearing 169 at one end thereof and pivotally afiixed to a third arm 172 at its second end by means of a pivot 173. The third arm 172 remote from the pivot 173 is affixed to the bearing 170 by means of a pivot 174 in the manner of the second and third arms 136 and 141 of FIGURE 3. A first lead screw 175 is disposed generally parallel to the first arm 166 and is in operative engagement with a nut 176 affixed to the bearing 169. A second lead screw 177 is disposed generally parallel to the first lead screw 175 and is in operative engagement with a nut 178 affixed to the bearing 170. The nuts 176 and 178 are in operative threaded engagement with the lead screws 175 and 177, respectively. The lead screws 175 and 177 have a generally like thread of like hand. The lead screw 175 is in operative association with a driving gear 179. The lead screw 177 is in operative association with a driving gear 180. The driving gears 179 and 180 in turn are in combination with a common gear 181 operated by a motor 182. The driving gear 179 has a smaller pitch diameter than the driving gear 180. Thus, for a given number of rotations of the gear 181, the bearing 169 is caused to travel a greater distance than the bearing 170.

In FIGURE 5 there is depicted the basic geometry controlling the operation of the apparatus in accordance with the present invention. The curve DD is a circular are representing the desired arc to be traced having radii r, r and AB. The line C'D represents the surface or reference line from which the arc is to be traced. The line CF is generally equivalent to the first arm of any of the embodiments of the invention and is pivoted at the point C. The triangles ABC' and the reference triangles ABC are equiangular as the line BC is maintained parallel to the radius r regardless of the angular disposition of the line CF. The lines AB and BC have constant length, therefore, for any position of C'F the following ratio holds:

wherein k is an arbitrary constant. As AB is constant so must AB remain constant and the point A will trace the arc DD. The point B traces an are having a center B and a radius having a length of BB". Point C traces an are having a center at B" and a radius of AB+BC.

In FIGURE 6, essentially the same basic geometry is employed. The line H'H represents the surface or reference line upon which an arc is to be constructed. G'J' and G'L are radii of the desired arc K. The generally parallel lines between H and J are equivalent to the first and second arms and racks 5t] and of the apparatus of FIGURES 1 and 2, respectively, and are treated as being colinear. The line GM :is pivoted at the point M and maintained parallel to the line G'L. The line 6] has a length of r/w wherein r is the length of the radius of the desired circle divided by a which is a constant greater than unity. The line 6] is parallel to G1. Therefore, the triangles G'H'J and the reference triangle GM] are equiangular and sides are proportional. The motion multiplying means indicated by azl ratio causes proportionality to be maintained between HT and JM. Thus, for any angular position of the line H'J as it is pivoted about the point H, the proportionality of the triangles remains constant and the radius r is maintained.

As is apparent from the foregoing specification, the present invention is susceptible of being embodied with various alterations and modifications which may differ particularly from those that have been described in the preceding specification and description. For this reason, it is to be fully understood that all of the foregoing is intended to be merely illustrative and is not to be construed or interpreted as being restrictive or otherwise limiting of the present invention, excepting as it is set forth and defined in the hereto-appended claims.

What is claimed is:

1. An apparatus for the generation of circular arcs which does not require access to the center of the arc, the apparatus comprising in cooperative combination a base,

a first arm having a first end and a second end, the second end of the first arm being pivotally attached to the base,

a second arm having a first and a second end slidably mounted on the first arm and adapted to move parallel thereto,

a third arm pivotally attached to the first arm, the third arm having a first end and a second end, the third arm being affixed to the first arm at the first end of the third arm,

a fourth arm having a first and a second end, the first end of the fourth arm being pivotally affixed to the third arm adjacent the second end thereof, the second end of the fourth arm being slidably attached to the first arm at a location disposed between the first end of the first arm,

a motion multiplying means operatively connected to the second end of the fourth arm and actuating the second arm, and

means to maintain the fourth arm parallel to the radius of the are being generated.

2. The apparatus of claim 1 wherein the fourth arm has a length between its points of pivotal attachment equal to r/a wherein r is the radius of the are being generated and a is the ratio of the motion multiplying means, a being greater than unity.

3. The apparatus of claim 1 wherein the motion multiplying means comprises a rack afiixed to the second end of the fourth arm,

a spur gear pivotally affixed to the first arm and in operative engagement withthe rack, and

a circular motion transmitting means coaxial with the spur gear and adapted to transmit linear motion to the second arm.

4. The apparatus of claim 1 wherein the base includes means to rotate the apparatus about a radius of the are being generated.

5. The apparatus of claim 1 wherein the base comprises a reference arm and a fifth arm having a first end and a second end, the fifth arm being pivotally affixed to the second end of the third arm by the first end and to the reference arm by the second end to provide in combination with the first, third, fifth and reference arms, a fourbar linkage forming a parallelogram.

6. The apparatus of claim 1 including a support pivotal- 1y afiixed to the end so constructed and arranged so as to permit rotation of the apparatus about a vertical radius of the circular arc to be generated and means to maintain the third arm in a vertical position.

7. The apparatus of claim 6 wherein the means to maintain the third arm in a vertical position comprises a reversible motor action by a gravity operated switch.

8. An apparatus for the generation of circular arcs without access to the center of the arc, the apparatus comprising in cooperative combination,

a first arm, the first arm having a first end and a second end, the second end of the first arm being pivotally afiixed to a base, the base adapted to be positioned on a radius of the desired arc,

a second arm, and

a third arm in cooperative adjustable combination with the first arm, the first, second and third arms defining a reference triangle,

means to maintain the second arm parallel to a radius of the desired circular are passing through the pivotal connection of the first arm to the base, and the third arm being disposed generally between the second arm and the second end of the first arm.

9. The apparatus of claim 8 wherein the third arm is slidably and pivotally aflixed to the first arm.

10. The apparatus of claim 9 wherein first and second arms are pivotally and slidably afiixed to the first arm.

11. The apparatus of claim 10 including means to position the second and third arms along the first arm wherein the ends of the second and third arms are closely adjacent when second and third arms are adjacent the base and are remotely positioned when the second and third arms are remote from the base.

12. The apparatus of claim 11 wherein the second and third arms which are in sliding pivotal connection to the base are in operative connection with a motion multiplying means.

13. The apparatus of claim 8 wherein the means to maintain the second arm in a vertical position comprises a reversible motor activated by a gravity operated switch.

14. The apparatus of claim 8 wherein a pivot is disposed in the base to permit movement of the first n of the first arm in a solid angle.

References Cited UNITED STATES PATENTS 2,996,920 8/1961 Mackenzie et al. 74-1 3,253,473 5/1966 Chisholm 741 XR 3,263,516 8/1966 Chisholm 74--1 XR FRED C. MATTERN, JR., Primary Examiner. C. J. HUSAR, Assistan! Examiner. 

